On approximations of eigenvalues and eigenvectors of perturbed linear operators (Q2713986)

Let \(L\) be a closed linear operator in a complex Hilbert space \(H\) and let \(B\) be a bounded linear operator. Given an isolated eigenvalue \(\lambda\), the corresponding eigenvector \(f\) of \(L\), and a number \(\varepsilon>0\), the problem is to find conditions on the operator \(B\) (the norm of \(B\) is small, etc.) which ensure that the operator \(L-B\) has an eigenvalue \(\lambda_1\) and an eigenvector \(f_1\), with \(|\lambda-\lambda_1|\leq \varepsilon\) and \(\|f-f_1\|\leq \varepsilon\). The author establishes exact estimates for the quantities \(|\lambda-\lambda_1|\) and \(\|f-f_1\|\) in terms of the norms of the operators \(P_iBP_j\) (\(i=1,2\)) with \(P_1\) the Riesz projection of \(L\) corresponding to the spectral set \(\{\lambda\}\) and \(P_2=I-P_1\). The existence of such an eigenvalue \(\lambda_1\) and an eigenvector \(f_1\) is also proven.